Proposition 4.4 ([1]}) 
Assume \(\varphi \in C^{0,\vartheta }(\mathbb {R})\)  with \(\vartheta \in (0,1]\) . If \(\rho \in L^1(\mathbb {R}^N)\cap L^\infty (\mathbb {R}^N)\) , then there exists a unique a.e.-solution \(T^{\textup {imp}}[\rho ]=w\in L^1(\mathbb {R}^N)\cap L^\infty (\mathbb {R}^N)\)  of (REF ). Moreover, for \(\rho ,\hat{\rho }\in L^1(\mathbb {R}^N)\cap L^\infty (\mathbb {R}^N)\) :
